Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $q = \dfrac{t + 1}{t} \div \dfrac{t + 1}{2} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{t + 1}{t} \times \dfrac{2}{t + 1} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (t + 1) \times 2 } { t \times (t + 1) } $ $ q = \dfrac {2 (t + 1)} {t (t + 1)} $ $ q = \dfrac{2(t + 1)}{t(t + 1)} $ We can cancel the $t + 1$ so long as $t + 1 \neq 0$ Therefore $t \neq -1$ $q = \dfrac{2 \cancel{(t + 1})}{t \cancel{(t + 1)}} = \dfrac{2}{t} $